Optimal. Leaf size=39 \[ -\frac {F\left (\cos ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right )|\frac {1}{42} \left (21+5 \sqrt {21}\right )\right )}{\sqrt [4]{21}} \]
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Rubi [A] time = 0.07, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1095, 420} \[ -\frac {F\left (\cos ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right )|\frac {1}{42} \left (21+5 \sqrt {21}\right )\right )}{\sqrt [4]{21}} \]
Antiderivative was successfully verified.
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Rule 420
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+5 x^2-x^4}} \, dx &=2 \int \frac {1}{\sqrt {5+\sqrt {21}-2 x^2} \sqrt {-5+\sqrt {21}+2 x^2}} \, dx\\ &=-\frac {F\left (\cos ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right )|\frac {1}{42} \left (21+5 \sqrt {21}\right )\right )}{\sqrt [4]{21}}\\ \end {align*}
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Mathematica [B] time = 0.11, size = 87, normalized size = 2.23 \[ \frac {\sqrt {-2 x^2-\sqrt {21}+5} \sqrt {\left (\sqrt {21}-5\right ) x^2+2} F\left (\sin ^{-1}\left (\sqrt {\frac {1}{2} \left (5+\sqrt {21}\right )} x\right )|\frac {23}{2}-\frac {5 \sqrt {21}}{2}\right )}{2 \sqrt {-x^4+5 x^2-1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 5 \, x^{2} - 1}}{x^{4} - 5 \, x^{2} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} + 5 \, x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 82, normalized size = 2.10 \[ \frac {\sqrt {-\left (\frac {5}{2}-\frac {\sqrt {21}}{2}\right ) x^{2}+1}\, \sqrt {-\left (\frac {5}{2}+\frac {\sqrt {21}}{2}\right ) x^{2}+1}\, \EllipticF \left (\left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ) x , \frac {5}{2}+\frac {\sqrt {21}}{2}\right )}{\left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-x^{4}+5 x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} + 5 \, x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {-x^4+5\,x^2-1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- x^{4} + 5 x^{2} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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